Statistical Methods for Psychology

5.3 Discrete versus Continuous Variables

In Chapter 1, a distinction was made between discrete and continuous variables. As mathe-

maticians view things, a discrete variable is one that can take on a countable number of dif-

ferent values, whereas a continuous variable is one that can take on an infinite number of

different values. For example, the number of people attending a specific movie theater

tonight is a discrete variable because we literally can count the number of people entering

the theater, and there is no such thing as a fractional person. However, the distance between

two people in a study of personal space is a continuous variable because the distance could

be 2, or 2.8, or 2.8173754814 feet. Although the distinction given here is technically cor-

rect, common usage is somewhat different.

In practice when we speak of a discrete variable, we usuallymean a variable that takes

on one of a relatively small number of possible values (e.g., a five-point scale of socioeco-

nomic status). A variable that can take on one of many possible values is generally treated

as a continuous variable if the values represent at least an ordinal scale. Thus we usually

treat an IQ score as a continuous variable, even though we recognize that IQ scores come

in whole units and we will not find someone with an IQ of 105.317. In Chapter 3, I referred

to the Achenbach Total Behavior Problem score as normally distributed, even though I

know that it can only take on positive values that are integers, whereas a normal distribu-

tion can take on all values between. I treat it as normal because it is close enough to

normal that my results will be reasonably accurate.

The distinction between discrete and continuous variables is reintroduced here because

the distributionsof the two kinds of variables are treated somewhat differently in probabil-

ity theory. With discrete variables we can speak of the probability of a specific outcome.

With continuous variables, on the other hand, we need to speak of the probability of ob-

taining a value that falls within a specific interval.

5.4 Probability Distributions for Discrete Variables

An interesting example of a discrete probability distribution is seen in Figure 5.1. The data

plotted in this figure come from a study by Campbell, Converse, and Rodgers (1976), in

which they asked 2164 respondents to rate on a 1–5 scale the importance they attach to var-

ious aspects of their lives (1 5 extremely important, 5 5 not at all important). Figure 5.1

6 q

118 Chapter 5 Basic Concepts of Probability

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0

Relative frequency of people

endorsing response

12345

Extremely

Health

Savings

Friends

Importance

Not at all

0

Figure 5.1 Distributions of importance ratings of three aspects of life